On 25 mar, 21:09, JT <jonas.thornv...@gmail.com> wrote:> On 25 mar, 14:07, David C. Ullrich <ullr...@math.okstate.edu> wrote:>>>>>>>> > On Sun, 24 Mar 2013 13:55:12 -0700 (PDT), Butch Malahide>> > <fred.gal...@gmail.com> wrote:> > >On Mar 24, 1:49 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:> > >> On Sun, 24 Mar 2013 10:37:46 -0700 (PDT), Butch Malahide> > >> > <fred.gal...@gmail.com> wrote:> > >> >On Mar 24, 10:53 am, David C. Ullrich <ullr...@math.okstate.edu>> > >> >wrote:> > >> >> [. . .]> > >> >> So the more interesting version of the question,> > >> >> in any case less trivial, amounts to this: Is there> > >> >> a measurable set D such that>> > >> >> 0 < m(D intersect I) < m(I)>> > >> >> for every open interval I,>> > >> >Didn't we just have that thread?>> > >> Yes.>> > >> >http://groups.google.com/group/sci.math/msg/0cfe35786f2279f0?hl=en>> > >> >> and such that m(D intersect [0,1]) = 1/2 ?>> > >> >OK, that's different.>> > >> Precisely! heh.>> > >But not so very different, is it?>> > I was joking. Of course once we know that there exists> > D with 0 < m(D intersect I) < m(I) for every interval I it> > follows that there exists such a D with the second condition.>> > By any of at least three arguments:>> > 1. The one I had in mind.>> > 2. The one you give below.>> > 3. By saying "Fine, now how in the world could> > it be that the value 1/2 is somehow excluded?">> > >Let D be a measurable set such that> > >0 < m(D intersect I) < m(I) for every interval I. It will suffice to> > >find an interval I such that m(D intersect I)/m(I) = 1/2. Since m(D) >> > >0, there is an interval J such that m(D intersect J)/m(J) > 1/2;> > >likewise, since m{R\D) > 0, there is an interval H such that m(D> > >intersect H)/m(H) < 1/2. Since the function f(a,b) = m(D intersect> > >(a,b))/(b-a) is continuous on the connected domain {(a,b): a < b},> > >there is an interval I = (a,b) such that f(a,b) = 1/2.>> I am not sure what you are onto here, but noone have forbidden> nullsamples, nor nullmembers of set for statistical use, not even> finitists.

There is empty boxes, empty samples, well there is actually quite alot of stuff that can have null magnitude, but there is not emptynumbers and not sets without members.